A Sobolev space theory for the stochastic partial differential equations with space-time non-local operators

• Published in: Journal of evolution equations 2022-09, Vol.22 (3), Article 57
• Main Authors: Kim, Kyeong-Hun, Park, Daehan, Ryu, Junhee
• Format: Article
• Language: English
• Subjects: Analysis; Brownian motion; Mathematics; Mathematics and Statistics; Operators (mathematics); Partial differential equations; Relativity; Sobolev space; White noise
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-022-00813-7

We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes ∂ t α u = ϕ ( Δ ) u + f ( u ) + ∂ t β ∑ k = 1 ∞ ∫ 0 t g k ( u ) d w s k , t > 0 , x ∈ R d as well as the SPDE driven by space-time white noise ∂ t α u = ϕ ( Δ ) u + f ( u ) + ∂ t β - 1 h ( u ) W ˙ , t > 0 , x ∈ R d . Here, α ∈ ( 0 , 1 ) , β < α + 1 / 2 , { w t k : k = 1 , 2 , … } is a family of independent one-dimensional Wiener processes and W ˙ is a space-time white noise defined on [ 0 , ∞ ) × R d . The time non-local operator ∂ t α denotes the Caputo fractional derivative of order α , the function ϕ is a Bernstein function, and the spatial non-local operator ϕ ( Δ ) is the integro-differential operator whose symbol is - ϕ ( | ξ | 2 ) . In other words, ϕ ( Δ ) is the infinitesimal generator of the d -dimensional subordinate Brownian motion. We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity results of solutions.